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3 votes
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130 views

A Talagrand inequality for the supremum of partial sums over function classes under dependence. (Reference request)

As a consequence to the Talagrand concentration inequality, it is well known that for a measurable space $(S,\mathcal{S})$ and an i.i.d. sample $X_1,...,X_n$ of $S$-valued random variables, if $\...
Daan's user avatar
  • 141
1 vote
0 answers
87 views

Supremum of sums of functions in $L^1$ taking random signs

Consider the Banach space $X=L^1([0,1])$, and let $n\gg1$ and $x_1, ..., x_n$ be any points in the unit sphere of $X$. Is there any reasonable lower bound for $$\sup_{(\epsilon_i)_{i=1}^n \in \{-1,+1\}...
HHN's user avatar
  • 393
2 votes
0 answers
75 views

Pullback by surjective submersion is injective?

Denote by $\mathcal{D}'_X$ the sheaf of distributions on a smooth manifold $X$. Let $M$ and $N$ be smooth manifolds and $\Phi: M \to N$ a submersion. Then $\Phi$ defines a unique morphism of sheaves $\...
psl2Z's user avatar
  • 261
-1 votes
1 answer
98 views

Spectrum of sum of positive and negative operators

Let $(\mathscr{H}, \langle \cdot, \cdot \rangle)$ be a separable complex Hilbert space, and let $\mathscr{D}$ be a dense subset of $\mathscr{H}$. Let $P: \mathscr{D} \to \mathscr{H}$ and $N: \mathscr{...
d'Alembert's user avatar
0 votes
0 answers
78 views

What properties does representing measure $\mu$ for $z\in\mathbb{D}^n$ has to satisfy so that $\nu=0$ is the only measure singular with respect to it?

Let $\mu$ and $\nu$ be two measures on the $\sigma$-Borel set $\mathcal{B}(\mathbb{D}^n)$. We say that $\mu$ is a representing measure for some point $z \in \mathbb{D}^n$, if $$\forall_{u \in A(\...
S-F's user avatar
  • 63
5 votes
2 answers
149 views

Showing an operator is (or not) closed on $L^2(\mathbb{R})$

I am linearizing nonlinear waves and get operators of the form below. Everything is considered in $L^2(\mathbb{R})$. Consider the operator $L_1=\frac{d}{dx}$. The domain is $H^1(\mathbb{R})$ and it is ...
Gateau au fromage's user avatar
2 votes
2 answers
105 views

Characterization of the dual of intersection of Banach spaces

Let $U,V$ Banach spaces and define $X = U\cap Y$ endowed with the norm $\|u\|_X = \|u\|_U + \|u\|_V$. If we take $\varphi \in U'$ and $\psi \in V'$, I can prove that $\varphi|_X + \psi|_X \in X'$, ...
Lucas Linhares's user avatar
1 vote
0 answers
98 views

Equivalence of Sobolev norms for smooth functions with compact support

Let $f\in C^\infty_c([0,1]^n)$, then we can extend it to a $1$-periodic smooth function $\tilde f$. We define the fourier transform (series) of $f$ ($\tilde f$):$$ \hat f(\xi):=\int e^{2\pi i x\cdot \...
Tian LAN's user avatar
  • 435
1 vote
1 answer
157 views

Is finding the CDF from the Laplace transform well-posed?

In my study of Dynamic Light Scattering, I came across the following inverse problem. Let $F(s):[0,T]\rightarrow[0,T]$ be the Laplace transform of a probability distribution $f(t)$ on the real line ...
Riemann's user avatar
  • 654
2 votes
0 answers
191 views

Smoothing property of the heat kernel on the one-dimensional torus

Let $G=G(x,t)$ be the heat kernel on the one-dimensional torus $\mathbb{T}^1,$ with $x \in \mathbb{T}^1$ and $t \in (0,T].$ $G$ is given by \begin{equation} G(x,t) = (4 \pi t)^{-1/2} \sum_{k \in \...
kumquat's user avatar
  • 185
0 votes
1 answer
231 views

Questions on the compactness of $L_1([0,1]^2)$'s unit sphere

Let $U$ denote the set of functions $f\in L_1([0,1]^2)$ such that $\int f=1$ and $f(x,y)\geq 0: a.e. (x,y)\in [0,1]^2$. Recently in my study I need to study the compactness of $U$. By Riesz's theorem ...
tom jerry's user avatar
  • 349
0 votes
1 answer
100 views

Embeddings of pseudo metric spaces into seminormed Spaces

There is a theorem stating that every metric space embeds isometrically into $\ell _{\infty}$. My question: is there a generalized result for pseudo metric spaces embedding isometrically into semi-...
DJ Forklift's user avatar
0 votes
1 answer
101 views

Limit sequence of regular function in $L_1$‘s unit sphere

Let $U$ denote the set of functions $f\in L_1([0,1]^2)$ such that $\int f=1$. For any $f\in U$, we say it is regular if $\int_{x_0\times [0,1]}f=\int_{[0,1]\times y_0}f=1$ for a.e. every $x_0, y_0\in [...
tom jerry's user avatar
  • 349
4 votes
0 answers
147 views

Weakly compact sets forced to contain $0$

Let $E$ be an infinite-dimensional real normed space and let $K\subset E$ be a weakly compact set such that, for each $\varphi\in E^*\setminus \{0\}$, there exists a unique $\tilde x\in K$ such that $$...
Biagio Ricceri's user avatar
0 votes
0 answers
121 views

How to find the inverse of this linear integral operator?

Let $f(x): \mathbb{R}^d \rightarrow \mathbb{R}$ be a function that decays ``fastly enough'' at infinity. We can define the following linear operator $$L[f](x):= \int_{\mathbb{R}^d} d^d y \, \frac{f(y)}...
mnerone's user avatar
3 votes
1 answer
158 views

Upper and lower bounds for a Rademacher-type expectation

Suppose that $\varepsilon_i$ are independent Rademacher random variables (that is, $ \mathbb{P}(\varepsilon_i=-1) = \mathbb{P}(\varepsilon_i=1) =1/2 $. Fix an $a\in\mathbb{R}^n$ and define the random ...
Aryeh Kontorovich's user avatar
5 votes
0 answers
160 views

Hartman uniform distribution of means

Background: for a discrete abelian group $G$, a character of $G$ is a homomorphism $\chi:G\to \mathbf S^1$, $\mathbf S^1$ being the circle group $\{z\in \mathbb C:|z|=1\}$ with ordinary multiplication....
John Griesmer's user avatar
-6 votes
1 answer
180 views

An analog of Anderson's result in C* algebra setting [closed]

Let $\mathcal{A}$ be a unital $C^{*}$-algebra and $S(\mathcal{A})$ denote the states space of $\mathcal{A}$. For $a\in \mathcal{A}$ , define $W(a) =\{\phi(a):\phi\in S(\mathcal{A})\}$ It's known that $...
SoG's user avatar
  • 307
2 votes
0 answers
179 views

Analytic continuation of $\int_V (f(x_1,\cdots,x_n))^s dx_i$

Let $V$ be an $n$-dimensional simplex, let $f(\boldsymbol{x}) = f(x_1,\cdots,x_n)\in \mathbb{C}[x_1,\cdots,x_n]$ be a product of linear polynomials that is non-zero in interior of $V$. Also let $E(\...
pisco's user avatar
  • 528
1 vote
1 answer
102 views

Does $C^{k,s-k}$ function with lipschitz lower order derivatives give a certain bound on the Taylor remainder?

Let $\Omega \subseteq \mathbb{R}$ be open (not necessarily an interval). Let $ s > 0$ and $k \in \mathbb{N}_0$ be such that $s \in (k, k+1]$. Suppose that $f \colon \Omega \to \mathbb{R}$ is an ...
Kacper Kurowski's user avatar
2 votes
1 answer
152 views

Co-locating slowly increasing smooth functions in two different ways

This question is subsequent from my previous one. I will write everything in detail for the sake of completeness. Let $g_1$ and $g_2$ be smooth functions on $\mathbb{R}$, whose derivatives of all ...
Isaac's user avatar
  • 3,477
5 votes
2 answers
256 views

On the closed convex hull of a weakly compact set

Let $H$ be an infinite-dimensional real Hilbert space and let $B$ be the closed unit ball of $H$. Let $K\subset B$ be a weakly compact set whose closed convex hull agrees with $B$. Question: does $K$ ...
Biagio Ricceri's user avatar
2 votes
0 answers
102 views

Semiclassical limit of spectral gap of Schrödinger operators with nonsmooth potential

Let $\Omega$ be a connected compact subset of $\mathbb{R}^d$. It is well known that for a smooth potential $V:\Omega \to \mathbb{R}$ that has a unique nondegenerate minimum $V(0) = 0$, the operator $H ...
Lwins's user avatar
  • 1,551
0 votes
0 answers
121 views

Is there a good or commonly accepted short notation for the set of differentiable, but not necessarily continuously differentiable maps?

Every once in a while I find myself in need of some short notation for the set of differentiable, but not continuously differentiable maps, say, $X \to Y$. Always having to specify "...
M.G.'s user avatar
  • 7,127
7 votes
0 answers
114 views

On the non-existence of a boundedly complete basic sequence

Is there a Banach space X with the property that neither X nor its dual X' has a boundedly complete basic sequence?
C Stuart's user avatar
3 votes
1 answer
182 views

Tensor product of a slowly increasing smooth function and a tempered distribution converging to a co-located product

Let $T$ be a tempered distribution on $\mathbb{R}$ and $g$ be a smooth function on $\mathbb{R}$ whose derivatives of all orders are all polynomially bounded (a.k.a. slowly increasing). For any pair of ...
Isaac's user avatar
  • 3,477
-1 votes
1 answer
168 views

Space of distributions on $[0,1]^2$: weakly compact or not?

Let $X_1,X_2$ be distributions on $[0,1]$ and let $X=(X_1,X_2)$ be the joint distribution of $X_1,X_2$. Let $\mathcal{X}$ be the set of all such joint distribution $X$. Question 1: Does $\mathcal{X}$ ...
tom jerry's user avatar
  • 349
1 vote
1 answer
40 views

Envelopes of functions with respect to some convex cone $\mathcal{F}$

Let's say we are given a function $f:\mathbb R ^d\to \mathbb R$ continuous. Assume that $\mathcal F$ is a convex cone of continuous functions ($\mathbb R^d$ to $\mathbb R$) closed under maxima. I am ...
J.R.'s user avatar
  • 291
2 votes
0 answers
75 views

Have the open Questions 1 and 2 from Section 7 of the paper "Integrals with values in Banach spaces" been answered?

Some context: I had previously asked the post below on MSE, but someone suggested I ask it here and delete the original post. In section 7 of the paper Integrals with values in Banach Spaces and ...
Sam's user avatar
  • 121
4 votes
0 answers
80 views

Interpolation-extrapolation scales of H. Amann

I am currently reading H. Amann's notes titled "Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems", and I have a question regarding the abstract ...
Michelangelo's user avatar
5 votes
1 answer
183 views

What is a natural interpretation of the commutator of the conditional expectation operator?

Notation: We denote by $\mathbb E_{\mathcal F} X$ the conditional expectation of the random variable $X$ with respect to the $\sigma$-algebra $\mathcal F$. Given two $\sigma$-algebras $\mathcal G, \...
Nate River's user avatar
  • 6,215
1 vote
1 answer
330 views

Does $\sum_{n=1}^{\infty}\frac{(-1)^n e^{\sin{n}}}{\sqrt{n}}$ converge?

I am trying to study the converge of the series $$\sum_{n=1}^{\infty}\frac{(-1)^n e^{\sin{n}}}{\sqrt{n}}$$ But $e^{\sin{n}}$ is not monotone, and the Abel's test rule fails here. Can someone help me? ...
pxchg1200's user avatar
  • 287
5 votes
1 answer
164 views

Does quadratic asymptotic growth imply log-Sobolev inequality?

Let $f : \mathbb{R}^n \rightarrow [0,\infty)$ be a smooth function and consider $h$ s.t $h(\vec{x}) = f(\vec{x}) + \lambda \Vert \vec{x} \Vert^2$. Does this imply that irrespective of any other ...
Student's user avatar
  • 617
10 votes
1 answer
314 views

Weakly metrizable sets in normed spaces

A similar question was asked on MSE without getting an answer. In the proof of lemma 1.2 of Asplund operators and holomorphic maps the author (my attempt to contact him failed because the only e-mail ...
Jochen Wengenroth's user avatar
0 votes
0 answers
60 views

Spectral analysis of Dirac operators coupled to gauge potential on $\mathbb{R}^n$

Dirac operators on compact manifolds seem to have been studied well, such as in this book and also this one. However, I cannot easily find comprehensive treatment of Dirac operators coupled to gauge ...
Isaac's user avatar
  • 3,477
3 votes
0 answers
196 views

Parabolic smoothing for semilinear PDE

Consider the semilinear energy-critical parabolic PDE in $\mathbb{R}^3$ \begin{align} \partial_t u &= \Delta u + |u|^{4/(n-2)}u = \Delta u + u^5\\ u(0,x) &= u_0\in \smash{\dot{H}}^1(\mathbb{R}^...
Student's user avatar
  • 537
0 votes
0 answers
78 views

Definition of Moore-Penrose inverse for unbounded self-adjoint operators?

I know there is a concept of Moore-Penrose or pseudoinverse of a matrix. I would like to know if one can define it for densely defined unbounded self-adjoint operators on Hilbert spaces as well. ...
InMathweTrust's user avatar
2 votes
1 answer
108 views

Separability is an interpolation property

I'm trying to prove that certain space, which can be obtained as an interpolation space, is separable. The fact that is separable is well known but i want to simplify it via interpolation. I haven't ...
Guillermo García Sáez's user avatar
0 votes
0 answers
45 views

Functional inequalities on neighbourhood graphs

Consider an open domain $\Omega \in \mathbb{R}^d$, say the unit disk in $\mathbb{R}^2$ with $N$ points sampled i.i.d. on it. One of the simplest possible (unnormalised) discrete Laplacian of a ...
Rundasice's user avatar
  • 111
6 votes
1 answer
170 views

Do projections in an $AW^\ast$-algebra form an orthomodular lattice?

I’m currently studying orthomodular lattices arising out of operator algebras. One of the most standard examples is the projection lattice of a von Neumann algebra - if $M$ acts on a Hilbert space $H$,...
David Gao's user avatar
  • 2,830
3 votes
1 answer
116 views

Does a bounded positive modular sesquilinear form on a $C^\ast$-algebra induces an element of its multiplier algebra?

This is a question that originates from my attempt at this question. Specifically, for a $C^\ast$-algebra $A$, I am attempting to construct a map $\phi: A \times A \to A$ s.t., $\phi$ is sesquilinear,...
David Gao's user avatar
  • 2,830
1 vote
1 answer
105 views

Constrained optimization over a set of functions

How to approach the following optimization problem: $$\text{minimize }\int_0^1 f(x) \, dx$$ over all (integrable) real-valued functions $f$ on $[0,1]$ satisfying $$1-x_1 x_2 \leq f(x_1)f(x_2)\text{ ...
Arkadi Predtetchinski's user avatar
3 votes
1 answer
109 views

Literature request: Covariance operators for Gaussian measures

I am looking to answer the question: If $\mathcal{B}$ is a separable Banach space and $R: \mathcal{B}^*\to\mathcal{B}$ is a symmetric and positive operator, then $\phi: \mathcal{B}^*\to\mathbb{R}, \...
ChocolateRain's user avatar
5 votes
1 answer
379 views

Nuclear vs Banach spaces: compactness properties

A question about the meaning from following excerpt from german wikipedia adressing interesting crucial feature of nuclear spaces opposing them from Banach spaces (transl.): While normed spaces, ...
user267839's user avatar
  • 6,028
2 votes
0 answers
47 views

Norm density of evaluation functionals in the space of weak$^*$ continuous multilinear functionals on products of dual Banach spaces

Let $K$ be a compact (metrizable) space and let $C(K)$ be the Banach space of continuous real-valued functions on $K$, equipped with the supremum norm. It is then well known that the dual space $C(K)^*...
kiliroy's user avatar
  • 56
0 votes
0 answers
141 views

The tensor product of two Fredholm operators

What can be said about the tensor product $T\otimes S$ of two Fredholm operators $T:X_1\to Y_1$ and $S:X_2 \to Y_2$ where $X_1,X_2,Y_1, Y_2$ are Banach spaces and tensor product of ...
Ali Taghavi's user avatar
1 vote
0 answers
40 views

relatively weakly compact sets in the projective tensor product of $\ell_p $ and a Banach space $X$

We will use the notation in [1]. A sequence $(x_n)$ in $X$ is called weakly $p$-summable ($p\ge 1$) if $(x^*(x_n))\in \ell_p$ for each $x^*\in X^*$. Equivalently, a sequence $(x_n)$ in $X$ is ...
Ioana Ghenciu's user avatar
4 votes
1 answer
180 views

Analytic function with values in $L^1$

Suppose that $(\Omega, \Sigma, \mu)$ is a measure space. Let $D$ be the unit open disk and $F : D \rightarrow L^1(\mu)$ be an analytic function. Is it true that for a.e. $w \in \Omega$ the function $F(...
Seven9's user avatar
  • 565
0 votes
0 answers
46 views

What's the problem in using spanning Bessel sequences that are not frames to decompose vectors?

This is related to a question I recently asked on math.SE. Consider a subset $G\equiv \{g_k\}_{k\in\mathbb{N} }\subseteq\mathcal H$ in a separable Hilbert space $\mathcal H$, and suppose $G$ spans the ...
glS's user avatar
  • 342
0 votes
0 answers
50 views

About extreme case on complex interpolation

I'm trying to prove some equality of spaces via complex interpolation with the usual Calderon functor $[,]_\theta$. If $(E_0,E_1)$ is a compatible couple, it is known that $$[E_0,E_1]_j, j=0,1,$$ is a ...
Guillermo García Sáez's user avatar